00:01
For this problem, we want to find the midpoint remand sum for the function f of x equals 4 plus cosine of pi x in the closed interval 05 for n equals 20 subintervals.
00:12
Now we recall the midpoint remand sum for the function y equals f of x for n sub intervals is given by the integral from a to b of f of x dx that's equal to the summation from k equals 1.
00:30
To n of f of ms .k times delta x, wherein delta x, this is equal to b minus a over n, and our m sub k are midpoints.
00:46
That's defined to be the sum of two consecutive numbers divided by two.
00:52
So that will be x of k minus 1 plus x and k all over 2.
01:00
So for the closed interval 05 and n equal to 20, we have delta x equal to 5 minus 0 over 20, that's equal to 1 over 4.
01:22
And since our m sub k equals x sub k minus 1 plus x sub k over 2, this tells us that our first midpoint m sum 1 is equal to x of 0 plus x sub 1 over 2.
01:42
And because x of 0 is the leftmost endpoint, that'll be 0 plus x sub 1 is just x 0 plus delta x all over 2.
01:56
This is just, you know, x of 0 is 0.
02:00
So that's just delta x over 2.
02:05
And because our subintervals are equal in length...